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There is a large consensus that gas in high-$z$ galaxies is highly turbulent,because of a combination of stellar feedback processes and gravitationalinstabilities driven by mergers and gas accretion. In this paper, we presentthe analysis of a sample of five Dusty Star Forming Galaxies (DSFGs) at $4\lesssim z\lesssim 5$. Taking advantage of the magnifying power of stronggravitational lensing, we quantified their kinematic and dynamical propertiesfrom ALMA observations of their [CII] emission line. We combined the dynamicalmeasurements obtained for these galaxies with those obtained from previousstudies to build the largest sample of $z \sim 4.5$ galaxies with high-qualitydata and sub-kpc spatial resolutions, so far. We found that all galaxies in thesample are dynamically cold, with rotation-to-random motion ratios, $V/\sigma$,between 7 to 15. The relation between their velocity dispersions and theirstar-formation rates indicates that stellar feedback is sufficient to sustainthe turbulence within these galaxies and no further mechanisms are needed. Inaddition, we performed a rotation curve decomposition to infer the relativecontribution of the baryonic (gas, stars) and dark matter components to thetotal gravitational potentials. This analysis allowed us to compare thestructural properties of the studied DSFGs with those of their descendants, thelocal early type galaxies. In particular, we found that five out of sixgalaxies of the sample show the dynamical signature of a bulge, indicating thatthe spheroidal component is already in place at $z \sim 4.5$.

Dynamical properties of z $\sim 4.5$ dusty star-forming galaxies and their connection with local early type galaxies